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با سلام و احترام🌹

به کانال پژوهشی و آموزشی رشته ریاضی خوش آمدید.

دوستان عزیز هدف از تشکیل این کانال موارد زیر می‌باشند:

✅ توانمندسازی و ترویج پژوهش و تفکر تحلیلی و علمی در رشته ریاضی

✅ توسعه دانش ریاضیات و ایجاد علاقه به رشته ریاضی در تمام سطوح آموزشی

✅ حمایت از دانشجویان و پژوهشگران در فعالیت‌های پژوهشی و آموزشی

✅ کاربردی کردن ریاضیات و توسعه خلاقیت جامعه علمی ریاضیات 

امید است که همه دوستان از مطالب مفید این کانال در راستای پیشرفت علمی، پژوهشی و آموزشی بهره‌مند شوند.

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#یادگیری_ریاضی_برتر

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ریاضیات، به عنوان پایه‌ای‌ترین علم، نقش اساسی در پیشرفت سایر رشته‌ها ایفا می‌کند. این رشته دارای گرایش‌های متنوعی است که هر کدام دنیایی از نظریه‌ها، کاربردها و چالش‌های منحصربه‌فرد را ارائه می‌دهند. آشنایی با این گرایش‌ها به دانشجویان رشته ریاضی کمک می‌کند تا بر اساس علایق و توانمندی‌های خود مسیر تحصیلی و شغلی و تخصص مناسب خود را بهتر انتخاب کنند. همچنین،  با توجه به اینکه ریاضیات زبان مشترک بسیاری از رشته‌ها است، آشنایی با گرایش‌های آن برای علاقه‌مندان به علوم دیگر نیز جذاب و کاربردی است.

❇️ توجه کنید که هر کسی چه بخواهد در حوزه نظری پژوهش کند و چه بخواهد به دنبال حل مسائل کاربردی باشد، ریاضیات همیشه جذاب و چالش‌برانگیز خواهد بود.

✅ در ادامه گرایش‌ها و زیرشاخه‌های مختلف رشته ریاضی را معرفی می‌کنیم و توضیحاتی در مورد آن گرایش بر اساس دانشگاه‌های معتبر آمریکا ارائه می‌دهیم.
اگر قصد دارید درک عمیق‌تری از گرایش‌های مختلف رشته ریاضی داشته باشید، می‌توانید مطالب تخصصی این کانال را در زمینه‌های پژوهشی و آموزشی دنبال کنید.

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#گرایش‌های_ریاضی

🌐 Algebra (جبر)

Algebra is the study of structures with operations resembling addition and/or multiplication, but more general than the usual number system of real numbers. The most important of these structures are groups, rings, and fields. Algebra refers to the use and manipulation of symbols, often with each representing some mathematical entity such as a quantity (think integer or real number), a set with a special structure (think group, ring, topological space, or vector bundle) or an element of such a set, or a relation (think function, partial order, or homomorphism). Manipulation of symbols usually follows specified rules that allow for operations such as addition, multiplication, composition, or action of one object upon another. In representation theory, for example, groups act on vector spaces; and in commutative algebra, elements of rings are viewed as functions on spaces.

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Algebra provides the mathematical tools to find unknown quantities from related known ones, the famous quadratic equation being a familiar example. The subject interacts with all of mathematics and many applied fields. For instance, the symmetries of pyramids, cubes, or indeed any object can be viewed through the lens of algebra. The seeds of modern algebra can be found in the Abel-Ruffini Theorem that no general formula for the zeroes of a polynomial equation of degree 5 is possible. The modern proof uses Galois Theory, which studies the zeroes of a polynomial by associating a group of symmetries.

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Because of its generality, algebra is used in many other fields of mathematics. Algebraic topology uses algebraic objects to study the properties of topological spaces. Algebraic number theory studies various rings that generalize the set of integers. Commutative ring theory underlies all of the structures of algebraic geometry. The symmetries of smooth manifolds form objects called Lie groups. The subject has many applications in the physical sciences, computer science, and engineering. For example, one important application of finite fields is the theory of error-correcting codes, which is used to transmit information between devices without data loss.

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#جبر

🌐 Combinatorics (ترکیبیات)

Combinatorics is the study of finite or discrete structures, such as networks, polyhedral, codes, or algorithms. The structures might have their origins in geometry, topology, computation, data analysis, probability, algebra, or natural sciences such as biology and physics. A hallmark of combinatorics is the breadth and depth of its interactions with other subjects. The overlap with algebra, for instance, is exemplified by number theory, which at its core concerns arithmetic (multiplicative or additive algebraic properties) of the integers (a countable discrete totally ordered set).

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It has played a particularly important role in the development of computer science. While it is arguably as old as counting, combinatorics has grown remarkably in the past half century alongside the rise of computers. The study of combinatorics involves general questions of enumeration and structure, matroid theory and aspects of graph theory, partially ordered sets, set partitions and permutations and combinatorial structures such as finite geometries and designs. Techniques tend to be algebraic and topological, involving methods from commutative ring theory, algebraic topology, representation theory, and Hopf algebras.

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Combinatorics borrows tools from diverse areas of mathematics. Examples include the probabilistic method, which was pioneered by Paul Erdös and uses probability to prove the existence of combinatorial structures with interesting properties, algebraic methods, such as the use of algebraic geometry to solve problems in discrete geometry and extremal graph theory, and topological methods beginning with Lovász' proof of the Kneser conjecture. A notable application in number theory is in the proof of the Green-Tao theorem that there are arbitrarily long arithmetic progressions of primes.

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#ترکیبیات

🌐 Number Theory (نظریه اعداد)

Number theory is concerned with the integers and rational numbers, and objects built out of them and related to them. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Integer or rational solutions of a system of polynomial equations in one or more unknowns is a part of number theory that goes by the name of Arithmetic Geometry. In Arithmetic Geometry too, certain power series, called Modular Forms, arise very naturally and play an important role in solving unfathomable mysteries. Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem”, which gives an asymptotic formula for the distribution of prime numbers. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis. In addition, conjectures in number theory have had an impressive track record of stimulating major advances outside the subject. For example, attempts to prove “Fermat’s Last Theorem” resulted in the development of large areas of algebra over the course of three centuries, and its recent proof involved a profound unifying force in modern mathematics known as the Langlands program.

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Gauss famously referred to the theory of numbers as the “Queen of Mathematics”. It is certainly one of the oldest and most beautiful branches of mathematics, and it is almost unique in its wide array of tools for subjects such as analysis, algebra, and topology. Many subfields of number theory include algebraic number theory, analytic number theory, p-adic arithmetic, p-adic cohomology, function field arithmetic, modular forms, the theory of L-functions, Diophantine geometry, and arithmetic dynamics. Number theory includes many famous questions, both solved and unsolved. For example, Fermat's Last Theorem (that there are no nontrivial integer solutions to x^n + y^n = z^n, with n > 2) is a famous result in number theory, due to Andrew Wiles. Famous open questions in number theory include the Birch and Swinnterton-Dyer conjecture, the Riemann Hypothesis, and Goldbach's conjecture.

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Contemporary number theory is developing rapidly through its interactions with many other areas of mathematics. Insights from ergodic theory have led to dramatic progress in old questions concerning the distribution of primes, geometric representation theory and deformation theory have led to new techniques for constructing Galois representations with prescribed properties, and the study of automorphic forms and special values of L-functions have been revolutionized by developments in both p-adic and arithmetic geometry as well as in pure representation theory. The ideas emerging from the Langlands Program (in its many modern guises) and from the developments that grew out of Wiles’ proof of Fermat’s Last Theorem continue to guide much of the ongoing research on the algebraic and geometric sides of the subject and in the analytic direction the synthesis of additive combinatorics and harmonic analysis continues to lead to breakthroughs in many directions. Modern era research topics like mathematics of encryption and theory of error correcting codes yield commonly used important real life applications of Algebra and Number Theory: computers, cell-phones, dvds, secured communication on internet all employ Algebra and Number Theory in substantial ways.

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#نظریه_اعداد

🌐 Representation Theory (نظریه نمایش)

Representation theory is fundamental in the study of objects with symmetry and a systematic study of how abstract groups (or algebras) can be represented by concrete linear transformations of a vector space. A guiding example is the symmetric group on four letters, which can be thought of as the rotational symmetries of a cube. Representation theory pervades diverse areas of mathematics and even particle physics and it seeks to understand all the possible ways an abstract collection of symmetries can arise. An early success was the work of Schur and Weyl, who computed the representation theory of the symmetric and unitary groups; the answer is closely related to the classical theory of symmetric functions, and a deeper study leads to intricate questions in combinatorics. In number theory, p-adic representation theory is central to the Langlands program, a family of conjectures that has guided a large part of number theory for the past forty years and the Langlands program posits a deep connection between representations of various Lie groups and representations of Galois groups, through the theory of L-functions.

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Nineteenth-century representation theory helped to explain the structure of electron orbitals, and 1920s representation theory is at the heart of quantum chromodynamics. The study of affine Lie algebras and quantum groups has brought many new ideas and viewpoints, and representation theory now furnishes a basic language for other fields, including the modern theory of automorphic forms. One fundamental problem involves describing all the irreducible unitary representations of each Lie group, the continuous symmetries of a finite-dimensional geometry. Doing so corresponds to identifying all finite-dimensional symmetries of a quantum-mechanical system. The representation theory of infinite-dimensional groups and supergroups is vital to string theory, statistical mechanics, integrable systems, tomography, and many other areas of mathematics and their applications.

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#نظریه_نمایش

🌐 Mathematical Logic (منطق ریاضی)

Mathematical logic is the study of the strengths and limitations of formal languages, proofs, and algorithms and their relationships to mathematical structures. It also aims to address foundational issues in mathematics and investigates the power of mathematical reasoning itself. The various subfields of this area are connected through their study of foundational notions: sets, proof, computation, and models. The period from the 1930s through the 1970s saw great progress in logic. Mathematical logic is a broad area encompassing proof theory, computability theory, set theory, and model theory. These areas are joined by their focus on the interplay between expressibility, definability and provability. Logic relates to theoretical computer science through computability theory and proof theory, to algebra, number theory, and algebraic geometry through model theory, and to analysis and ergodic theory through set theory and infinite combinatorics.

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The exciting and active areas of logic today are set theory, model theory, and connections with computer science. Set theory addresses various ways to axiomatize mathematics, with implications for understanding the properties of sets having large infinite cardinalities and connections with the axiomatization of mathematics. Model theory investigates particular mathematical theories, such as complex algebraic geometry, and has been used to settle open questions in these areas. Theoretical computer science developed partially out of logic and Computational complexity, as part of theoretical computer science, is deeply connected to questions in proof theory and computability theory as well as to related areas including combinatorics and probability. The core goal of computational complexity is to determine the limits of computation; this includes some of the most fundamental open questions in mathematics and theoretical computer science, including the P versus NP question.

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#منطق_ریاضی

🌐 Topology (توپولوژی)

Topology is concerned with the intrinsic properties of the shapes of spaces and it studies properties of spaces that are invariant under deformations. Topology is the qualitative study of shapes and spaces by identifying and analyzing features that are unchanged when the object is continuously deformed. The modern field of topology draws from a diverse collection of core areas of mathematics. Much of basic topology is most profitably described in the language of algebra i.e. groups, rings, modules, and exact sequences. General topology now forms one of the foundations of modern analysis and geometry. The notion of moduli space was invented by Riemann in the 19th century to encode how Riemann surfaces vary in families; today, the study of geometric and homotopy-theoretic aspects of moduli spaces is an important subject with strong ties to algebraic and symplectic geometry. It also leads to interesting dynamical systems and group theory. More algebraic aspects of topology study homotopy theory and algebraic K-theory, and their applications to geometry and number theory. Topology itself has many active branches, many of which are intimately connected to other areas of mathematics and science: knot theory (biology, physics, algebra), algebraic topology (differential geometry, algebra, data analysis, physics, chemistry, engineering, robotics), geometric group theory (algebra).

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A “topology” is precisely the minimum structure on a set that allows one to even define what “continuous” means. Continuity, which refers to changes that may “stretch” or “fold” but never “tear”, is a fundamental concept in mathematics and science. By establishing the “minimal” requirements to understand continuity, topology has applications in almost every branch of mathematics and science, such as analysis (analytical constructions such as differential forms play a crucial role in topology), differential geometry and partial differential equations (through the modern subject of gauge theory), algebraic geometry (for instance, through the topology of algebraic varieties), combinatorics (knot theory), and theoretical physics (general relativity and the shape of the universe, string theory). In addition, topology can strikingly be used to study a wide variety of more "applied" areas, ranging from the structure of large data sets to the geometry of DNA.

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One class of spaces which plays a central role in mathematics, and whose topology is extensively studied, are the n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space. The case of manifolds of dimension n=1 is straightforward, and the case where n=2 was understood thoroughly in the 19th century. Moreover, intense activity in the 1960’s (including the pioneering work of Browder, Milnor, Novikov, and Smale) expresses the topology of manifolds of dimension n>4 in terms of an elaborate but purely algebraic denoscription. The study of manifolds of dimension n=3 and 4 is quite different from the higher-dimensional cases; and, though both cases n=3 and 4 are quite different in their overall character, both are generally referred to as low-dimensional topology. Low-dimensional topology is currently a very active part of mathematics, benefiting greatly from its interactions with the fields of partial differential equations, differential geometry, algebraic geometry, modern physics, representation theory, number theory, and algebra.

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#توپولوژی

For the case of manifolds of dimension n=3, a conjectural classification picture emerged in the 1970s, thanks to the work of William Thurston, in terms of symmetric geometries. Specifically, Thurston conjectured that every three-manifold can be decomposed canonically into pieces, each of which can be endowed with one of eight possible geometries. Elements of this vast picture are presently unfolding thanks to the Ricci Flow equations introduced by Richard Hamilton, which have been used by Grigory Perelman to solve the century-old Poincaré conjecture, and have also shed light on Thurston’s more general geometrization conjecture. The central role of Thurston’s conjecture in three-manifold topology has helped place hyperbolic geometry, the richest of the eight geometries, into the research forefront.

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The case of manifolds of dimension n=4 remains the most elusive. In view of the foundational results of Freedman, understanding manifolds up to their topological equivalence is a theory which is similar in character to the higher-dimensional manifold theory. However, the theory of differentiable four-manifolds is quite different. The subject was fundamentally transformed by the pioneering work of Simon Donaldson, who was studying moduli spaces of solutions to certain partial differential equations which came from mathematical physics. Studying algebro-topological properties of these moduli spaces, Donaldson came up with very interesting smooth invariants for four-manifolds which demonstrated the unique and elusive character of smooth four-manifold topology. In the case where the underlying manifold is Kähler, these moduli spaces also admit an interpretation in terms of stable bundles, and hence shed light on the differential topology of smooth algebraic surfaces. Since Donaldson’s work, the physicists Seiberg and Witten introduced another smooth invariant of four-manifolds. Since then, the study of four-manifolds and their invariants has undergone several further exciting developments, tying them deeply with ideas from symplectic geometry and pseudo-holomorphic curves, and hence forming further bridges with algebraic and symplectic geometry, but also connecting them more closely with knot theory and three-manifold topology.

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#توپولوژی

🌐 Algebraic Topology (توپولوژی جبری)

Algebraic topology is the interplay between “continuous” and “discrete” mathematics. Continuous mathematics is formulated in its general form in the language of topological spaces and continuous maps. Discrete mathematics is used to express the concepts of algebra and combinatorics. The notion of shape is fundamental in mathematics. Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus. Algebraic methods become important in topology when working in many dimensions, and increasingly sophisticated parts of algebra are now being employed. In algebraic topology, we investigate spaces by mapping them to algebraic objects such as groups, and thereby bring into play new methods and intuitions from algebra to answer topological questions. For example, the arithmetic of elliptic curves, which was at the heart of Andrew Wiles' solution of the Fermat conjecture, has been lifted into topology, giving new and very powerful tools for the study of geometric objects.

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Classical algebraic topology consists of the construction and use of functors from some category of topological spaces into an algebraic category, say of groups. But one can also postulate that global qualitative geometry is itself of an algebraic nature. Consequently there are two important view points from which one can study algebraic topology: homology and homotopy. Homology, invented by Henri Poincaré, is without doubt one of the most ingenious and influential inventions in mathematics. The basic idea of homology is that we start with a geometric object (a space) which is given by combinatorial data (a simplicial complex). Then the linear algebra and boundary relations determined by these data are used to produce homology groups. Homotopy theory begins again with the construction of functors from topology to algebra. But this approach is important from another view point. Homotopy theory shows that the category of topological spaces has itself a kind of (hidden) algebraic structure. The development of algebraic topology led to the development of many of the key concepts of algebra, including homological algebra, category theory, Lie groups/algebras, and K-theory.

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#توپولوژی_جبری